8 Interaction of Harmonic Waves of Different Types …
119
Medium displacements, summed up with ingoing wave displacements, are equal
to the displacement of the first plate bearing layer. The displacement of the second
medium is the same as the displacement of the second plate bearing layer. Shear
stresses are assumed to be vanishing.
8.7 Fourier Decomposition of Unknown Functions
All unknown function, as well for the plate as for both media, are decomposed
into trigonometric series and satisfy the boundary conditions of the simple support
(Sheddon 1951).
K =
∞
n=1
w
(1)
0n , w
(2)
0n , w
(1)
n , w
(2)
n , p n , p 1n , p 2n , w ∗n , σ 33∗n , σ
(1)
33n , σ
(2)
33n ,
ϕ
(l)
n , ψ
(l)
1n , ε
(l)
11n , ε
(l)
33n , σ
(l)
11n , σ
(l)
13n , σ
(l)
33n , w an , w cn
sin(λx),
L =
∞
n=1
u
(1)
0n , u
(2)
0n , u
(1)
n , u
(2)
n , ψ
(l)
3nm , ε
(l)
13n , u
a
1n , q
1
n
cos(λx),
λ =
π n
l
.
(8.30)
Dynamic equations for the plate are represented hence through the Fourier coefficients. Then, taking into account (8.30), the equation system for the plate (8.14) can
be solved, and for the normal displacements on the surfaces, we have the formulae
(8.31).
− λ
2
n Bu
a
1n + ω
2
ρ a u
a
1n + 2q
1
n = 0,
− Dλ
4
n w cn + ω
2
ρ c w cn − 2λ n k 1 q
1
n + p 1n − p 2n = 0,
− Dλ
4
n w an +
ω
2
ρ aw − 2c 3
w an + p 1n + p 2n = 0,
u
a
1n − k 1 λ n w cn + k 2 λ
2
n q
1
n + k 3 q
1
n = 0.
(8.31)
ε
(l)
11n = −λ n u
(l)
1n , ε
(l)
33n =
∂w
(l)
n
∂z
,
ε
(l)
13nm =
1
2
∂u
(l)
1nm
∂z
+ λ 1n w
(l)
nm
,
θ
(l)
n = −λ n u
(l)
1m +
∂w
(l)
n
∂z
(n ≥ 1).
(8.32)
σ
(l)
11n = λθ
(l)
n + 2με
(l)
11n , σ
(l)
33n = λθ
(l)
n + 2με
(l)
33n ,
σ
(l)
13n = 2με
(l)
13n .
(8.33)